Multiple solutions in the calculation of axisymmetric contraction flow of an upper convected maxwell fluid

Abstract

The failure of finite-element techniques to generate steady two-dimensional solutions for the flow of an upper convected Maxwell (UCM) fluid into a 4: 1 axisymmetric, sudden contraction for large values of Deborah number, De, is found to stem from the existence of multiple solutions to the algebraic equations. The presence of a limit point De L, that is inherent in the mathematical problem rather than present as a numerical artifact, is demonstrated by a sequence of two-dimensional calculations that cover a wide range of approximation refinement. Refinement is achieved by decreasing mesh size, by increasing the polynomial order (linear to quadratic) in the Lagrangian functions interpolating stress, and by allowing slip along the solid walls to reduce the stress gradients that have proven difficult to approximate in the past. With the quadratic stress interpolation (QQL) method, increasing refinement leads to an unequivocal De L between 0.6 and 0.8 with the exact value depending on whether or not slip is included. Stress interpolation using bilinear polynomials (QLL) results in highly oscillatory stress and velocity fields for Deborah numbers below a limiting value. These oscillations and the behavior of De L with mesh refinement suggest that solution structure for the QLL method does not reflect the behavior of the original differential equations.